MIT OpenCourseWare http ocw mit edu 18 02 Multivariable Calculus Fall 2007 For information about citing these materials or our Terms of Use visit http ocw mit edu terms 18 02 Practice Exam 2 A Problem 1 10 points 5 5 Let f x y xy x4 a Find the gradient of f at P 1 1 b Give an approximate formula telling how small changes x and y produce a small change w in the value of w f x y at the point x y 1 1 Problem 2 20 points On the topographical map below the level curves for the height function h x y are marked in feet adjacent level curves represent a di erence of 100 feet in height A scale is given a Estimate to the nearest 1 the value at the point P of the directional derivative u is the unit vector in the direction of dh ds where u h h 0 and 0 Estimate to the nearest 1 b Mark on the map a point Q at which h 2200 x y h the value of at Q y 2200 2100 P 2000 1900 1000 Problem 3 10 points Find the equation of the tangent plane to the surface x3 y z 2 3 at the point 1 1 2 Problem 4 20 points 5 5 5 5 A rectangular box is placed in the rst octant as shown with one corner at the origin and the three adjacent faces in the coordinate planes The opposite point P x y z is constrained to lie on the paraboloid x2 y 2 z 1 Which P gives the box of greatest volume z a Show that the problem leads one to maximize f x y xy x3 y xy 3 and write down the equations for the critical points of f P y b Find a critical point of f which lies in the rst quadrant x 0 y 0 c Determine the nature of this critical point by using the second derivative test x d Find the maximum of f in the rst quadrant justify your answer Problem 5 15 points In Problem 4 above instead of substituting for z one could also use Lagrange multipliers to maximize the volume V xyz with the same constraint x2 y 2 z 1 a Write down the Lagrange multiplier equations for this problem b Solve the equations still assuming x 0 y 0 Problem 6 10 points Let w f u v where u xy and v x y Using the chain rule express x y fu and fv w w and in terms of x y Problem 7 15 points Suppose that x2 y xz 2 5 and let w it numerically when x y z 1 1 2 x3 y Express w z as a function of x y z and evaluate y
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